Exercise
A
monopolist
faces the demand curve
Q = 120  3P
. Find the revenue maximizing price and quantity:
Maximizing revenue is
equivalent to maximizing profit when MC = 0.
By the (Pchoke + MC)/2 rule, the optimal price is 20. Then, the optimal quantity is 60.
Exercise
Find the profits at the
Nash equilibrium
. Consider two firms that play a
simultaneous move Cournot competition game
.
The market inverse demand is given by
P = 120 – Q
. The marginal cost is zero for both firms: P1 = 120 – Q1 – Q2 … MR1 = 120 – Q1
– 2(Q2) = 0 … solve for Q1, then set Q1 and Q2 = Qne, solve … obtain Qne and Pne, solve for profits. Profits = (Pne – MC) x Qne
Exercise
Consider now a
Cournot
game where each firm is allowed to choose either the Nash equilibrium quantity or half the
monopoly quantity, only. Draw the normal form of the simultaneous move game. By the Pchoke/MC rule the monopoly profit
maximizing price is 60, which makes the monopoly quantity 60. If each firm produces half of that, each makes a profit of 1800. If one
firm produces half the monopoly quantity and the other produces the Nash equilibrium outcome, the price is 50: (P = 120 – Qnash 
.5Qmonop) The profits are 1500 and 2000, respectively: Nash= (500)(40) .5 Monop= (500)(30)
Nash
Half Monopoly
Nash
1600, 1600 1500, 2000
Half Monopoly
2000, 1500 1800, 1800
Exercise
Argue why the game above is a prisoner’s dilemma: Each firm has a dominant strategy. Also, the dominant strategy
equilibrium is pareto inferior to (1800, 1800).
Exercise
A
monopolist
produces at constant marginal cost and faces two
segmented markets
with the following demand curves:
Qa
= 100 P … Qb = 100 – 2(P)
In which market will he charge a higher price if he maximizes profits? :
E = P/(Pchoke  P)
The first
demand has a higher choke price and is therefore more inelastic for any price.
By the mark up elasticity formula:
the lower the
elasticity, the higher the price you can charge, w/o the quantity changing that much. Therefore, the price should be higher in segment A.
Exercise
Assume that you estimate the
elasticity of demand
for luxury cars to be 0.8. On the basis of this information, can you
conclude that at least one car firm is not maximizing profits? Explain: It is true that any firm that maximizes profits will produce at the
elastic (E > 1) part of the demand. However, the estimate of 0.8 refers to the industry demand…isn’t informative about the pricing
behavior of individual firms.
Exercise
Consider the
Stackelberg
(
sequential
) version of a
twoplayer Cournot competition
game. Would you rather be the
follower, or the leader? (Assume that both firms have the same constant marginal cost.): We know that in the
Stackelberg
version of
the
Cournot
game, the following statements hold, relative to the Nash equilibrium outcome of the simultaneous move game: The
leader wants the follower to decrease his quantity, because the game has negative externalities. The leader achieves this by
increasing his quantity, because actions are strategic substitutes. This lowers the profits of the follower, because the game has
negative externalities. The profit of the leader goes up in all games. Therefore, the leader has higher profits than the follower and you
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 Spring '06
 DonaldBrown
 Supply And Demand, Nash, inverse demand

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